Riesz multiplier convergent spaces of operator valued series and a version of Orlicz- Pettis theorem
aa r X i v : . [ m a t h . F A ] F e b RIESZ MULTIPLIER CONVERGENT SPACES OF OPERATOR VALUED SERIESAND A VERSION OF ORLICZ- PETTIS THEOREM
MAHMUT KARAKUŞ AND RAMAZAN KAMA
Abstract.
It is not usual to characterize an operator valued series via completeness of multiplierspaces. In this study, by using a series of bounded linear operators, we introduce the space M ∞ R (cid:0) P k T k (cid:1) of Riesz summability which is a generalization of the Cesàro summability. Therefore, we give thecompleteness criteria of these spaces with c ( X ) -multiplier convergent operator series. It is a naturalconsequence that one can characterize the completeness of a normed space through M ∞ R (cid:0) P k T k (cid:1) which will be assumed that is complete for every c ( X ) -multiplier Cauchy operator series. Then,we characterize the continuity and the (weakly) compactness of the summing operator S from themultiplier space M ∞ R (cid:0) P k T k (cid:1) to an arbitrary normed space Y through c ( X ) -multiplier Cauchy and ℓ ∞ ( X ) -multiplier convergent series, respectively. We also prove that if P k T k is ℓ ∞ ( X ) -multiplierCauchy, then the multiplier space of weakly Riesz-convergence associated to the operator valued series M ∞ wR (cid:0) P k T k (cid:1) is subspace of M ∞ R (cid:0) P k T k (cid:1) . Among other results, finally, we obtain a new version ofthe well-known Orlicz-Pettis theorem by using Riesz-summability. Introduction
Let N be the set of positive integers, R and C also be the real and complex fields as usual,respectively. By ω , we denote the space of all real (or complex) valued sequences and any vectorsubspace of ω is also called as a sequence space . A K space is a locally convex sequence space X containing φ on which coordinate functionals π k ( x ) = x k are continuous for every k ∈ N , the set ofpositive integers, where φ is the space of finitely non-zero sequences spanned by the set { e k : k ∈ N } . e k is the sequence whose only non-zero term is 1 in the k th place for all k ∈ N , and e is also thesequence with e = (1 , , ... ) . A complete linear metric (or complete normed) K space is called an F K (or a BK ) space. Let X ⊃ φ be a BK space and x = ( x k ) ∈ X . Then, by x [ n ] = P nk =1 x k e k for all n ∈ N , we denote the n th section of x . It is said that X ⊃ φ is an AK space if (cid:13)(cid:13) x [ n ] − x (cid:13)(cid:13) X → , as n → ∞ , for each x = ( x k ) ∈ X . The spaces ℓ ∞ , c and c of bounded, convergent and null sequencesare BK spaces, respectively, with the sup norm k x k ∞ = sup k ∈ N | x k | .Let X , Y be any two sequence spaces and A = ( a nk ) be an infinite matrix of complex numbers a nk , where k, n ∈ N . Then, we say that A defines a matrix transformation from X into Y and wedenote it by writing A : X → Y, if for every sequence x = ( x k ) ∈ X the sequence Ax = { ( Ax ) n } , the A -transform of x , is in Y ; where ( Ax ) n = X k a nk x k . (1.1)For simplicity in notation, here and after, the summation without limits runs from 1 to ∞ . By ( X : Y ) ,we denote the class of all matrices A such that A : X → Y . Thus, A ∈ ( X : Y ) if and only if theseries on the right side of (1.1) converges for each n ∈ N . Furthermore, the sequence x is said to be A -summable to a ∈ C if Ax converges to a which is called the A -limit of x . The reader can refer to[6, 7] and [26] for recent results and related topics in summability.We say that the series P k x k in a normed space X is unconditionally convergent ( uc ) or uncon-ditionally Cauchy ( uC ) if the series P k x π ( k ) converges or is a Cauchy series for every permutation π of N . It is called weakly unconditionally Cauchy ( wuC ) if for every permutation π of N , the se-quence (cid:0) P nk =1 x π ( k ) (cid:1) is a weakly Cauchy sequence or as a useful result, P k x k is wuC if and only if P k | x ∗ ( x k ) | < ∞ for all x ∗ ∈ X ∗ , the space of all linear and bounded (continuous) functionals definedon X . It is well known that every wuC series in a Banach space X is uc if and only if X contains no Mathematics Subject Classification.
Key words and phrases.
Riesz summability; operator valued series; multiplier convergent series; summing operator;Orlicz-Pettis Theorem. opy of c ; (cf. [4, pp. 42, 44], [8, p. 44] and [19, p. 18]). The reader can refer to the Diestel’ s famousmonograph [8] given on the theory of sequences and series in Banach spaces, Albiac and Kalton [4]for specific investigations of Banach spaces and Marti’s [19] for basic sequences and fundamentals ofbases.Let X and Y be two normed spaces and ω ( X ) be the space of all X -valued sequences. By ℓ ∞ ( X ) , c ( X ) and c ( X ) , we denote the spaces of all X -valued bounded, convergent and null sequences,respectively. φ ( X ) also denotes the space of X -valued finitely non-zero sequences. Let V be a vectorspace of X -valued sequences equipped with a locally convex Hausdorff topology. If the maps x =( x k ) x k from V into X are continuous for all k ∈ N , then V is called as a K space. If x ∈ X , thenby e k ⊗ x , we denote the sequence whose only non-zero term is x in the k th place for all k ∈ N . If φ ( X ) ⊂ V and T k ∈ B ( X : Y ) , the space of all bounded and linear operators defined from X into Y forall k ∈ N , then we say that the series P k T k is V -multiplier convergent or V -multiplier Cauchy if theseries P k T k x k converges in Y or is a Cauchy series in Y , i.e., the partial sums of the series P k T k x k form a norm Cauchy sequence in Y for all x = ( x k ) ∈ V . The reader may refer to [22] for the vectorvalued multiplier spaces associated with operator valued series (OVS) and more detailed informationon multiplier convergent series.The multiplier form of a series P k x k in a Banach space X associated with an arbitrary real orcomplex sequence a = ( a k ) is given as P k a k x k and is also important to understand the behaviorof the series P k x k in X . The series P k x k in X is wuC (or uc ) series if and only if P k a k x k isconvergent for every null (or bounded) sequence a = ( a k ) , that is, P k x k is a c -(or an ℓ ∞ -) multiplierconvergent series. As another example, a series P k x k is subseries convergent if and only if it is m -multiplier convergent series, where m is the space of all finite range sequences. Let also recall that aseries P k x k is subseries convergent if and only if P k x n k is convergent for all subsequences ( n k ) of N .There is also an important result on subseries convergence which states that in a normed space X , asubseries convergent series is ℓ ∞ -multiplier Cauchy, if also X is sequentially complete then the seriesis ℓ ∞ -multiplier convergent.The studies of characterizations of Banach spaces, and obtaining new multiplier spaces by meansof summability methods have been an interesting field of research in the theory of modern analysis.In [20], McArthur investigated some closed linear subspaces of the space of the weakly unconditionalsummable sequences in a Banach space X and gave some inclusion relations between these spaces.One can refer to the Aizpuru et al. [1, 3], Pérez-Fernández et al. [21], Kama and Altay [10], Kamaet al. [11], Kama [9, 12], Karakuş [13], Karakuş and Başar [14, 15, 16] and Swartz [22, 23] for recentstudies on multiplier convergent series.Through the rest of the paper, we deal with the Riesz transformation which is a generalization ofCesàro mean C of order one given as C : ω −→ ωx = ( x k ) C x = (cid:0) n P nk =1 x k (cid:1) n ∈ N . (1.2)A series P k x k in a normed space X is said to be Cesàro or weak Cesàro convergent to x ∈ X and isdenoted by C − P k x k = x or wC − P k x k = x , if lim n →∞ " n n X k =1 ( n − k + 1) x k = x or lim n →∞ " n n X k =1 ( n − k + 1) x ∗ ( x k ) = x ∗ ( x ) for all x ∗ ∈ X ∗ . Riesz transformation is also given by Rx = R n n X k =1 r k x k ! n ∈ N or any sequence x = ( x k ) k ∈ N ∈ ω , where r = ( r k ) is a sequence of nonnegative reals with r > suchthat R n = n X k =1 r k with lim n →∞ R n = ∞ . (1.3)The Riesz method is regular under the condition (1.3) and is reduced for r = e to the Cesàro meanof order one. The corresponding matrix R = ( r nk ) to the Riesz method with respect to the sequence r = ( r k ) can be given, as follows; r nk := (cid:26) r k R n , ≤ k ≤ n, , k > n for all k, n ∈ N .In this study, our main purpose is to give some results related to the some new multiplier by usingRiesz summability method as a generalization of Cesàro summability, and construct some new classesof sequence spaces associated to an OVS in a Banach space having similar properties with the classesdefined in [5, 13, 15, 16] and [23]. Among other results, we give some new characterizations of c ( X ) -and ℓ ∞ ( X ) -multiplier convergent series through this method.The following lemma states a classical result of the wuC series in a normed space X , and givesan example of multiplier convergent series which characterizes unconditionally convergent series andweakly unconditionally Cauchy series with ℓ ∞ -multiplier convergent series and c -multiplier convergentseries, respectively; [8, p. 44]. Lemma 1.1.
The following statements hold: (a)
In a normed space X , a formal series P n x n is a wuC series if and only if there exists apositive real H such that H = sup n ∈ N ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 a k x k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) : | a k | ≤ , k = 1 , , . . . , n ) . (b) A formal series P n x n in a Banach space X is uc (respectively wuC ) series if and only iffor any ( t n ) ∈ ℓ ∞ (respectively for any ( t n ) ∈ c ), P n t n x n converges, that is, P n x n is an ℓ ∞ -(respectively a c -) multiplier convergent series. Vector Valued Multiplier Spaces Through Riesz Summability and compactsumming operator
In this section, we give some definitions on Riesz convergent or Riesz summable sequences in areal normed space.
Definition 2.1.
A sequence x = ( x k ) in a real normed space X is said to be Riesz convergent ( R -convergent) and weakly Riesz convergent ( wR -convergent) to x ∈ X which is called the R -limit and wR -limit of x , and is denoted by R − lim k →∞ x k = x and wR − lim k →∞ x k = x , if lim n →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R n n X k =1 r k x k − x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 0 and lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R n n X k =1 r k x ∗ ( x k ) − x ∗ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 for all x ∗ ∈ X ∗ hold, respectively. Riesz summability of a sequence x = ( x k ) (or R - and wR -convergence of a series) in a real normedspace X is also given, as follows: efinition 2.2. A series P k x k in a real normed space X is said to be Riesz convergent ( R -convergent)and weakly Riesz convergent ( wR -convergent) to x ∈ X which is called the R -sum and wR -sum of x ,and is denoted by R − P k x k = x and wR − P k x k = x , if lim n →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R n n X k =1 r k s k − x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 0 and lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R n n X k =1 r k x ∗ ( s k ) − x ∗ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 for all x ∗ ∈ X ∗ hold, respectively. Here, s k = P kj =1 x j is the sequence of partial sums of the series P k x k . In the following, we introduce the spaces of R -convergence and weakly R -convergence of multiplierswith association of an OVS, and obtain some new characterizations of c ( X ) - and ℓ ∞ ( X ) -multiplierconvergent (Cauchy) series. We also present the necessary and sufficient conditions for the continuityand (weak) compactness of the summing operator S defined from these multiplier spaces to anothernormed space Y , and derive some results by the previous works [5, 15] and [23]. Definition 2.3.
Suppose that X and Y are two normed spaces and T k ∈ B ( X : Y ) for all k ∈ N . Wedefine the space M ∞ R (cid:0) P k T k (cid:1) of R -convergence associated to the P k T k as; M ∞ R (cid:0) X k T k (cid:1) := ( x = ( x k ) ∈ ℓ ∞ ( X ) : X k T k x k is R -convergent ) , (2.1) and endowed with the sup norm. It can be easily checked that the inclusions φ ( X ) ⊆ M ∞ R (cid:0) X k T k (cid:1) ⊆ ℓ ∞ ( X ) (2.2)hold.Firstly, we give the following theorem related to the completeness of the normed space M ∞ R (cid:0) P k T k (cid:1) through c ( X ) -multiplier convergence. Theorem 2.4.
Regarding two Banach spaces X and Y , and a series P k T k , where T k ∈ B ( X : Y ) forall k ∈ N , the following assertions are equivalent: (i) The series P k T k is c ( X ) -multiplier convergent. (ii) The space M ∞ R (cid:0) P k T k (cid:1) is a Banach space.Proof. (i) ⇒ (ii): Let us suppose that the series P k T k is c ( X ) -multiplier convergent. Then, from Part(a) of Lemma 1.1 there exists H > such that H = sup n ∈ N ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 T k x k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) : k x k k ≤ for all k ∈ { , , . . . , n } ) . Now suppose that ( x m ) = ( x mk ) k ∈ N is a Cauchy sequence in M ∞ R (cid:0) P k T k (cid:1) . Then, since ℓ ∞ ( X ) isa Banach space and the inclusion relation (2.2) holds, one can find x = ( x k ) ∈ ℓ ∞ ( X ) such that x m → x , as m → ∞ . Now, we shall prove that x ∈ M ∞ R (cid:0) P k T k (cid:1) . For an arbitrary ǫ > , we have m ∈ N satisfying k x m − x k < ǫ H for every m > m . Now, it is immediate by easy calculation that Hǫ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R n n X k =1 r k k X j =1 T j ( x mj − x j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ H or every m > m and k ∈ N . Therefore, for every ǫ > there exists m ∈ N such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R n n X k =1 r k k X j =1 T j ( x mj − x j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ǫ for all m > m and k ∈ N . Since ( x m ) is a Cauchy sequence in M ∞ R (cid:0) P k T k (cid:1) , we have a sequence ( y m ) in the Banach space Y for which the following inequality (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R n n X k =1 r k k X j =1 T j ( x mj − y m ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ǫ holds for every n > n . Then, since Y is a Banach space and ( y m ) is a Cauchy sequence we can find y ∈ Y such that y m → y , as m → ∞ , and so for every ǫ > , k y m − y k < ǫ holds. Finally, for every ǫ > there exists n ∈ N such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R n n X k =1 r k k X j =1 T j x j − y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R n n X k =1 r k k X j =1 T j ( x j − x mj ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ++ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R n n X k =1 r k k X j =1 T j x mj − y m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + k y m − y k < ǫ holds for every n > n . Hence, x ∈ M ∞ R (cid:0) P k x k (cid:1) .(ii) ⇒ (i): Let us suppose that the multiplier space M ∞ R (cid:0) P k T k (cid:1) is complete and let us take x =( x k ) ∈ c ( X ) . Then, we have c ( X ) ⊆ M ∞ R ( P k T k ) since the space M ∞ R ( P k T k ) is closed and theinclusion φ ( X ) ⊂ M ∞ R ( P k T k ) holds. Therefore, the series P k T k x k is R -convergent for all x = ( x k ) ∈ c ( X ) . From the monotonicity of c ( X ) , the series P k T k x k is subseries R -convergent, and so is weaklysubseries R -convergent. From a version of Orlicz-Pettis theorem for regular matrices [3, Theorem 4.1], P k T k x k is subseries norm convergent, that is, the series P k T k is ℓ ∞ ( X ) -multiplier convergent, andso is c ( X ) -multiplier convergent. (cid:3) Corollary 2.5.
Let X and Y be any given two Banach spaces and a series P k T k with T k ∈ B ( X : Y ) for all k ∈ N . Then, the following statements are equivalent: (i) The series P k T k is c ( X ) -multiplier convergent. (ii) The inclusion c ( X ) ⊆ M ∞ R (cid:0) P k T k (cid:1) holds. Remark 2.6.
Let X and Y be any given two Banach spaces, and T k ∈ B ( X : Y ) for all k ∈ N . Themultiplier spaces M ∞ (cid:0) P k T k (cid:1) , M ∞ C (cid:0) P k T k (cid:1) and M ∞ f (cid:0) P k T k (cid:1) are introduced in [23] , [5] and [15] ,respectively, as follows; M ∞ (cid:0) X k T k (cid:1) := ( x = ( x k ) ∈ ℓ ∞ ( X ) : X k T k x k is convergent ) , (2.3) M ∞ C (cid:0) X k T k (cid:1) := ( x = ( x k ) ∈ ℓ ∞ ( X ) : X k T k x k is Cesàro convergent ) , (2.4) M ∞ f (cid:0) X k T k (cid:1) := ( x = ( x k ) ∈ ℓ ∞ ( X ) : X k T k x k is almost convergent ) . (2.5) Now, by considering the definitions of the multiplier spaces M ∞ R (cid:0) P k T k (cid:1) , M ∞ (cid:0) P k T k (cid:1) , M ∞ C (cid:0) P k T k (cid:1) and M ∞ f (cid:0) P k T k (cid:1) respectively given by (2.1), (2.3), (2.4)and (2.5), we have the following inclusions: M ∞ (cid:0) X k T k (cid:1) ⊆ M ∞ f (cid:0) X k T k (cid:1) ⊆ M ∞ C ( X k T k ) ⊆ M ∞ R ( X k T k ) . y bearing in mind Theorem 2.4 together with previous results given in [5] and [15], we have thefollowing: Corollary 2.7.
Let X and Y be Banach spaces, and T k ∈ B ( X : Y ) for all k ∈ N . Then, the followingassertions are equivalent: (i) The series P k T k is c ( X ) -multiplier convergent. (ii) M ∞ f (cid:0) P k T k (cid:1) is a Banach space. (iii) M ∞ C (cid:0) P k T k (cid:1) is a Banach space. (iv) M ∞ R (cid:0) P k T k (cid:1) is a Banach space. Remark 2.8.
Let X and Y be normed spaces and T k ∈ B ( X : Y ) for all k ∈ N . Consider the vectorvalued multiplier Cauchy space CM ∞ (cid:0) P k T k (cid:1) and the vector valued multiplier Riesz-Cauchy space CM ∞ R (cid:0) P k T k (cid:1) which are associated to an OVS, and defined by CM ∞ (cid:0) X k T k (cid:1) := ( x = ( x k ) ∈ ℓ ∞ ( X ) : X k T k x k is a Cauchy series ) (2.6) and CM ∞ R (cid:0) X k T k (cid:1) := ( x = ( x k ) ∈ ℓ ∞ ( X ) : X k T k x k is a Riesz-Cauchy series ) . (2.7)By using the spaces given in (2.6) and (2.7), we have the following corollary as an analogue ofcorresponding result given in [1, Remark 2.4]. Corollary 2.9.
Let X and Y be normed spaces and T k ∈ B ( X : Y ) for all k ∈ N . The followingstatements are equivalent: (i) P k T k is c ( X ) -multiplier convergent. (ii) CM ∞ (cid:0) P k T k (cid:1) is a Banach space. (iii) CM ∞ R (cid:0) P k T k (cid:1) is a Banach space. The following theorem is an analogue of Theorem 2.5 given in [5], and characterizes the complete-ness of a normed space by the vector valued multiplier space M ∞ R (cid:0) P k T k (cid:1) . Since the proof is similarto the case M ∞ C (cid:0) P k T k (cid:1) given in [5], we omit details; (see also [23, Corollary 1.8]). Theorem 2.10.
Let X and Y be given any normed spaces such that X is complete and T k ∈ B ( X : Y ) for all k ∈ N . Then, the space Y is also complete if and only if M ∞ R (cid:0) P k T k (cid:1) is complete for every c ( X ) -multiplier Cauchy series. In the following theorem, we give a new characterization of continuity of the certain summingoperator by using Riesz summability method and c ( X ) -multiplier Cauchy series. Theorem 2.11.
Let X and Y be any normed spaces and T k ∈ B ( X : Y ) for all k ∈ N . Then, thesumming operator S defined by S : M ∞ R (cid:0) P k T k (cid:1) −→ Yx = ( x k ) x = R − P k T k x k (2.8) is continuous if and only if the series P k T k is c ( X ) -multiplier Cauchy.Proof. Let us suppose that S is continuous and define the set G by G := ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 T k x k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) : k x k k ≤ for all k ∈ { , , . . . , n } ) . (2.9)Since the inclusion φ ( X ) ⊂ M ∞ R ( P k T k ) holds, the series P k T k is c ( X ) -multiplier Cauchy from theinequality H = sup n ∈ N G ≤ kSk .Conversely, let us suppose that P k T k is c ( X ) -multiplier Cauchy series. Therefore, the set G defined by (2.9) is bounded (see [23, Theorem 1.3]) and so, H = sup n ∈ N G . If x = ( x k ) ∈ M ∞ R (cid:0) P k T k (cid:1) ,then the proof follows from the inequality kS x k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R − X k T k x k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ H k x k . Combining Theorem 2.11 with the results due to Altay and Kama [5] and Karakuş and Başar [15];we have the following:
Corollary 2.12.
Let X , Y be any normed spaces and T k ∈ B ( X : Y ) for all k ∈ N . Then, thefollowing statements are equivalent: (i) The series P k T k is c ( X ) -multiplier Cauchy. (ii) S : M ∞ f (cid:0) P k T k (cid:1) → Y is continuous. (iii) S : M ∞ C (cid:0) P k T k (cid:1) → Y is continuous. (iv) S : M ∞ R (cid:0) P k T k (cid:1) → Y is continuous. Now, we give the characterization of the (weakly) compactness for the summing operator S byusing Riesz summability method and ℓ ∞ ( X ) -multiplier convergent series. Let us note that X need notto be complete. Theorem 2.13.
Let X be any normed space, Y be a Banach space and T k ∈ B ( X : Y ) for all k ∈ N .Then the series P k T k is ℓ ∞ ( X ) -multiplier convergent if and only if the summing operator S definedby (2.8) is compact (weakly compact).Proof. Let us suppose that S is compact. If x = ( x k ) ∈ ℓ ∞ ( X ) , then the set H := (X k ∈F e k ⊗ x k |F finite and k x k k ≤ ) ⊂ M ∞ R (cid:0) X k T k (cid:1) is bounded. By the hypothesis, S ( H ) := ( R − X k ∈F T k x k |F finite and k x k k ≤ ) is relatively compact. Therefore, the series P k T k x k is subseries norm R -convergent, and so is weaklysubseries R -convergent [22, Theorem 2.48]. Further, by a consequence of Orlicz-Pettis theorem forregular matrices [3], the series P k T k x k is subseries norm convergent; that is the series P k T k is ℓ ∞ ( X ) -multiplier convergent.Conversely, suppose that the series P k T k is ℓ ∞ ( X ) -multiplier convergent. We define the operators S Rn by S Rn : M ∞ R (cid:0) P k T k (cid:1) −→ Yx = ( x k ) Rn ( x ) = R − P nk =1 T k x k for all n ∈ N . It is sufficient to prove that kS Rn − Sk → , as n → ∞ . Since P k T k is ℓ ∞ ( X ) -multiplierconvergent, then the series P k T k x k is uniformly R -convergent for k x k k ≤ [22, Corollary 11.11].Therefore, lim n →∞ (cid:13)(cid:13) S Rn − S (cid:13)(cid:13) = lim n →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R − n X k =1 T k x k ! − R − X k T k x k !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = lim n →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R − ∞ X k = n +1 T k x k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 0 with k x k k ≤ . This step completes the proof. (cid:3) Combining the related results in [5] and [15] with Theorem 2.13, we have the following:
Corollary 2.14.
Let X be any normed space, Y be a Banach space and T k ∈ B ( X : Y ) for all k ∈ N .Then, the following statements are equivalent: (i) The series P k T k is ℓ ∞ ( X ) -multiplier convergent. (ii) S : M ∞ f ( P k T k ) → Y is compact (weakly compact). (iii) S : M ∞ C ( P k T k ) → Y is compact (weakly compact). (iv) S : M ∞ R (cid:0) P k T k (cid:1) → Y is compact (weakly compact). ow, we may introduce the multiplier space of weak R -convergence associated to the series P k T k and obtain the corresponding results similar to the previous theorems and corollaries. Definition 2.15.
Let X and Y be normed spaces, and T k ∈ B ( X : Y ) for all k ∈ N . The vector valuedmultiplier space M ∞ wR (cid:0) P k T k (cid:1) of weakly R -convergence associated to the series P k T k is defined by M ∞ wR (cid:0) X k T k (cid:1) := ( x = ( x k ) ∈ ℓ ∞ ( X ) : X k T k x k is wR -convergent ) and endowed with the sup norm. Since the inclusion M ∞ R (cid:0) P k T k (cid:1) ⊆ M ∞ wR ( P k T k ) clearly holds, we have the following inclusionswhich are similar to the relation in (2.2): φ ( X ) ⊆ M ∞ R (cid:0) X k T k (cid:1) ⊆ M ∞ wR (cid:0) X k T k (cid:1) ⊆ ℓ ∞ ( X ) . (2.10)Since the completeness of the multiplier space M ∞ wR (cid:0) P k x k (cid:1) is given by means of c ( X ) -multiplierconvergent series by using the similar technique for proving the completeness of M ∞ R (cid:0) P k x k (cid:1) , we omitthe proof of following theorem. Theorem 2.16.
Let X and Y be any given Banach spaces, and T k ∈ B ( X : Y ) for all k ∈ N . Then,the series P k T k is c ( X ) -multiplier convergent if and only if M ∞ wR (cid:0) P k T k (cid:1) is a Banach space. Corollary 2.17.
Let X and Y be Banach spaces, and T k ∈ B ( X : Y ) for all k ∈ N . Then, the series P k T k is c ( X ) -multiplier convergent if and only if the inclusion c ( X ) ⊆ M ∞ wR (cid:0) P k T k (cid:1) holds. Remark 2.18.
Let us suppose that X and Y are Banach spaces and T k ∈ B ( X : Y ) for all k ∈ N .Then, the multiplier space M ∞ w (cid:0) P k T k (cid:1) is introduced in [23] as M ∞ w (cid:0) X k T k (cid:1) := ( x = ( x k ) ∈ ℓ ∞ ( X ) : X k T k x k is weakly convergent ) . Now, if the series P k T k is a c ( X ) -multiplier convergent, then the series P k y ∗ ( T k x k ) is convergentfor all x = ( x k ) ∈ c ( X ) and for all y ∗ ∈ Y ∗ , that is, the series is weakly convergent. It is known byCorollary 2.5 that x = ( x k ) ∈ M ∞ R (cid:0) P k T k (cid:1) , and so x = ( x k ) ∈ M ∞ wR (cid:0) P k T k (cid:1) . This means that thereexists y ∈ Y with wR − P k T k x k = y such that X k y ∗ ( T k x k ) = R − X k y ∗ ( T k x k ) = y ∗ ( y ) . Therefore, the inclusion M ∞ R (cid:0) P k T k (cid:1) ⊆ M ∞ w ( P k T k ) holds. However, we have no an idea on thesufficient conditions for the reverse inclusion. By combining the previous results and Theorem 2.16, we derive the following for the analogue ofCorollary 2.7 in the weak topology:
Corollary 2.19.
Let X and Y be Banach spaces, and T k ∈ B ( X : Y ) for all k ∈ N . Then, thefollowing assertions are equivalent: (i) The series P k T k is c ( X ) -multiplier convergent. (ii) M ∞ wf (cid:0) P k T k (cid:1) is a Banach space. (iii) M ∞ wC (cid:0) P k T k (cid:1) is a Banach space. (vi) M ∞ wR (cid:0) P k T k (cid:1) is a Banach space. Following theorem is the analogue of Theorem 3.3 of [5]. Since the proof is similar to the case M ∞ wC (cid:0) P k T k (cid:1) , we omit details. Theorem 2.20.
Let X be a Banach space, Y be any normed space and T k ∈ B ( X : Y ) for all k ∈ N . Then, Y is complete if and only if the multiplier space M ∞ wR (cid:0) P k T k (cid:1) is complete for every c ( X ) -multiplier Cauchy series. t is well-known that if Y is a Banach space, then B ( X : Y ) is a Banach space. So, Theorem 2.20and also Theorem 2.10 can be used for proving completeness of B ( X : Y ) .Now, we give the following theorem which characterizes the continuity of weak summing operatorwith c ( X ) -multiplier Cauchy series. Theorem 2.21.
Let X and Y be normed spaces, and T k ∈ B ( X : Y ) for all k ∈ N . Then, the summingoperator S defined by S : M ∞ wR (cid:0) P k T k (cid:1) −→ Yx = ( x k ) x = wR − P k T k x k . (2.11) is continuous if and only if the series P k T k is c ( X ) -multiplier Cauchy.Proof. Let us suppose that the summing operator S defined by (2.11) is continuous and consider theset G given by (2.9). Then, the desired result follows from the inequality sup n ∈ N G = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) wR − X k T k x k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ kSk , since the inclusion φ ⊂ M ∞ wR (cid:0) P k T k (cid:1) holds.Conversely, if P k T k is c ( X ) -multiplier Cauchy series, then the set G is bounded (see [23, Theorem1.3]) and so H = sup n ∈ N G . If x = ( x k ) ∈ M ∞ wR (cid:0) P k T k (cid:1) , then the proof follows from the inequality kS x k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R − X k y ∗ ( T k x k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ H k x k for all y ∗ ∈ B Y . (cid:3) From Theorem 2.21 and the conclusions due to Altay and Kama [5] and Karakuş and Başar [15],we have the following:
Corollary 2.22.
Let X and Y be two normed spaces, and T k ∈ B ( X : Y ) for all k ∈ N . Then, thefollowing statements are equivalent: (i) The series P k T k is c ( X ) -multiplier Cauchy. (ii) S : M ∞ wf (cid:0) P k T k (cid:1) → Y is continuous. (iii) S : M ∞ wC (cid:0) P k T k (cid:1) → Y is continuous. (iv) S : M ∞ wR (cid:0) P k T k (cid:1) → Y is continuous. Theorem 2.23.
Let X be any normed space, Y be a Banach space and T k ∈ B ( X : Y ) for all k ∈ N .Then, the series P k T k is ℓ ∞ ( X ) -multiplier convergent if and only if the summing operator S definedby (2.11) is compact (weakly compact).Proof. Since the proof can be given by the similar way used in proving Theorem 2.13, we omit details. (cid:3)
By Theorem 2.23 and the results in [5] and [15], we have the following:
Corollary 2.24.
Let us suppose that X and Y are any normed spaces such that Y is complete, and T k ∈ B ( X : Y ) for all k ∈ N . Then the following statements are equivalent: (i) The series P k T k is ℓ ∞ ( X ) -multiplier convergent. (ii) S : M ∞ wf ( P k T k ) → Y is compact (weakly compact). (iii) S : M ∞ wC ( P k T k ) → Y is compact (weakly compact). (iv) S : M ∞ wR (cid:0) P k T k (cid:1) → Y is compact (weakly compact). Prior to passing to the next section, we present Proposition 2.25 which states, besides the inclusiongiven in (2.10), the iclusion M ∞ wR (cid:0) P k T k (cid:1) ⊆ M ∞ R (cid:0) P k T k (cid:1) also holds. Proposition 2.25.
Let X and Y be normed spaces. If P k T k is ℓ ∞ ( X ) -multiplier Cauchy, then M ∞ wR (cid:0) P k T k (cid:1) = M ∞ R (cid:0) P k T k (cid:1) . roof. Let X and Y be normed spaces, and x = ( x k ) ∈ M ∞ wR (cid:0) P k T k (cid:1) . Then, there exists y ∈ Y suchthat R − P k y ∗ ( T k x k ) = y ∗ ( y ) for every y ∗ ∈ Y ∗ . From hypothesis, since the partial sums of the series P k T k x k form a Cauchy sequence in Y , there exists y ∗∗ ∈ Y ∗∗ such that R − P k T k x k = y ∗∗ . If weconsider the uniqueness of the limit, then we have y ∗∗ = y . Therefore, x = ( x k ) ∈ M ∞ R (cid:0) P k T k (cid:1) . (cid:3) A version of Orlicz-Pettis Theorem for Riesz summability
The Orlicz-Pettis Theorem is one of the important results in the Theory of Functional Analysis.Many generalizations and applications of this theorem can be found in [5, 17, 18, 22, 24, 25, 27].Before stating and proving the Orlicz-Pettis Theorem by means of Riesz convergence, we will give thefollowing definition (see [22]):
Definition 3.1.
The space λ has the infinite gliding hump property ( ∞ -GHP) if whenever x ∈ λ and { σ m } is an increasing sequence of intervals, there exist a subsequence { p m } and t p m > , t p m → ∞ such that every subsequence of { p m } has a further subsequence { q m } such that the coordinatewise sumof the series P m t q m χ σ qm x ∈ λ . In the following, we give a new version of Orlicz-Pettis theorem for vector valued multiplier Rieszconvergent space of OVS by using Antosik-Mikusinski matrix theorem.
Theorem 3.2.
Let λ have ∞ -GHP and T k ∈ B ( X : Y ) for all k ∈ N . If the series P k T k is λ − multiplier Riesz convergent with respect to weak topology of Y , then the series P k T k is λ − multiplierRiesz convergent with respect to strong topology of Y .Proof. Let ǫ > . If the conclusion is false, there exists x ∈ λ , ( y ∗ n ) ⊂ Y ∗ bounded sequence and anincreasing sequence of intervals { σ n } such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R − X k ∈ σ n y ∗ n ( T k x k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > ǫ (3.1)for all n ∈ N . Since λ has ∞ -GHP, there exist a subsequence { p n } and t p n > , t p n → ∞ suchthat every subsequence of { p n } has a further subsequence { q n } such that P n t q n χ σ qn x ∈ λ . Now, weconsider the matrix H = [ h ij ] defined by h ij = X m ∈ σ pj y ∗ p i t p i ( T m ( t p j x m )) for all i, j ∈ N . Since ( y ∗ i ) is bounded and t p i → ∞ , the columns of H converge to 0. On the otherhand, since λ has ∞ -GHP, we have P n t q j χ σ qj x ∈ λ for every subsequence { q j } and hence the subseries X j X m ∈ σ qj T m ( t q j x m ) is weakly Riesz convergent. Then, we have lim i →∞ X j a iq j = lim i →∞ X j X m ∈ σ qj y ∗ p i t p i ( T m ( t q j x m )) . From Antosik-Mikusinski theorem ([22] Appendix D), the matrix H is Riesz convergent to zero asdiagonal, which is contradiction with (3.1). (cid:3) Conclusion
Pérez-Fernández et al. [21] gave some properties of a normed space X like completeness andbarrelledness through the wuC series in X and weakly* unconditionally Cauchy series in X ∗ . Aizpuruet al. [1] gave a new characterization of wuC and uc series through Cesàro summability methodand studied the new spaces associated to the series in a Banach space for proving completeness andbarrelledness of normed spaces, and obtained a new version of Orlicz-Pettis theorem in scalar case.They also characterized the wuC series by means of continuity of a linear mapping from these newspaces to a normed space. Aizpuru et al. [2] gave a new characterization of wuC and uc series viathe completeness of some new subspaces of ℓ ∞ obtained from almost convergence and also presented new version of Orlicz-Pettis theorem. In [3], they also generalized the results given in [1] to anyregular matrix. In recent times, Karakuş and Başar [14] introduced a slight generalization of almostconvergence and gave some new multiplier spaces associated to the series P k x k in a normed space bymeans of this new summability method. Swartz constructed some new versions of the Orlicz-Pettistheorem for multiplier convergent series under continuity assumptions on some linear operators, andgave applications to spaces of continuous linear operators, [24, 25]. In [27], Yuanghong and Rongluproved that multiplier convergence of OVS is closely related to the AK property of the sequence spaces,and obtained corresponding versions of Orlicz-Pettis theorems. Kama and Altay [10] obtained somenew multiplier space by using Fibonacci sequence spaces, and Kama et al. [11] gave similar results formultiplier spaces which were obtained from backward difference matrix. Most recently, León-Saavedraet al. [17, 18] also obtained some new versions of Orlicz-Pettis theorem by using w p -summability anda general summability methods.Swartz [23] extended these studies (which are concerned with the scalar valued multiplier spaces)to the case of operator valued series and vector valued multipliers. Later, Altay and Kama [5] intro-duced the vector valued multiplier spaces by means of Cesàro convergence and a sequence of continuouslinear operators. Karakuş gave the vector valued multiplier spaces S Λ ( T ) and S w Λ ( T ) by means of Λ -convergence and a series of bounded linear operators together with the characterization the com-pleteness of both of the normed spaces S Λ ( T ) and S w Λ ( T ) via c ( X ) -multiplier convergence of anoperator series, [13]. Quite recently, Karakuş and Başar [15, 16] and Kama [9] introduced the vec-tor valued multiplier spaces of almost summability, a generalized almost summability and statisticalCesàro summability, respectively.It is worth noting to the reader that one can obtain corresponding results of the present paper byusing any regular matrix A = ( α nk ) n,k ∈ N instead of the Riesz matrix. References [1] Aizpuru, A., Gutiérrez-Dávila, A., Sala A.: Unconditionally Cauchy series and Cesàro summability. J. Math. Anal.Appl. , 39-48 (2006)[2] Aizpuru, A., Armario, R., Pérez-Fernández, F. J.: Almost summability and unconditionally Cauchy series. Bull.Belg. Math. Soc. Simon Stevin , 635-644 (2008)[3] Aizpuru, A., Pérez-Eslava, C., Seoane-Sepúlveda, J. B.: Matrix summability methods and weakly unconditionallyCauchy series. Rocky Mountain J. Math. (2), 367-380 (2009)[4] Albiac, F., Kalton, N. J.: Topics in Banach Space Theory. 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(2), 1-11 (2019)[14] Karakuş, M., Başar, F.: A generalization of almost convergence, completeness of some normed spaces with wuC series and a version of Orlicz-Pettis theorem. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM (4), 3461-3475 (2019)[15] Karakuş, M. Başar, F.: Operator valued series, almost summability of vector valued multipliers and (weak) com-pactness of summing operator, J. Math. Anal. Appl. (1), 1-16 (2020)[16] Karakuş, M. Başar, F.: Vector valued multiplier spaces of f λ -summability, completeness through c ( X ) -multiplierconvergence and continuity and compactness of summing operators, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. AMat. RACSAM 114, 169(2020)[17] León-Saavedra, F., Moreno-Pulido, S., Sala, A.: Orlicz-Pettis Type Theorems via Strong p-Cesàro Convergence.Numer. Funct. Anal. Optim. (7), 798-802 (2019)[18] León-Saavedra, F., de la Rosa, R., del Pilar, M., Sala, A.: Orlicz-Pettis Theorem through Summability Methods.Mathematics, (10), 895 (2019)[19] Marti, J.T.: Introduction to the Theory of Bases. Springer-Verlag, Berlin · Heidelberg (1969)[20] McArthur, C. W.: On relationships amongst certain spaces of sequences in an arbitrary Banach space. Canad. J.Math. , 192-197 (1956)
21] Pérez-Fernández, F. J., Benítez-Trujillo, F. Aizpuru, A.: Characterizations of completeness of normed spaces throughweakly unconditionally Cauchy series. Czechoslovak Math. J. (125), 889-896 (2000)[22] Swartz, C.: Multiplier Convergent Series. World Scientific Publishing, Singapore (2009)[23] Swartz, C.: Operator valued series and vector valued multiplier spaces. Casp. J. Math. Sci. (2), 277-288 (2014)[24] Swartz, C.: A bilinear Orlicz-Pettis theorem. J. Math. Anal. Appl. (1): 332-337 (2010)[25] Swartz, C.: The Orlicz-Pettis theorem for multiplier convergent series. In Advanced Courses of MathematicalAnalysis V. Hackensack, NJ: World Science Publication, 295-306 (2016).[26] Wilansky, A.: Summability through Functional Analysis. North-Holland Mathematics Studies , Amsterdam · New York · Oxford (1984)[27] Yuanhong T, Ronglu L.: Orlicz-Pettis theorem for λ − multiplier convergent operator series. Bull. Aust. Math. Soc. , 247-252 (2007)(Mahmut Karakuş) Van Yüzüncü Yıl University, Faculty of Science, Department of Mathematics,65100 - Van, Turkey
Email address : [email protected], [email protected] (Ramazan Kama) Siirt University, Faculty of Education, Department of Mathematics and PhysicalSciences Education, 56100 - Siirt, Turkey
Email address : [email protected]@siirt.edu.tr